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By modular invariance, the dominant contribution in the T2 -- 0 region must go likeZUv ~ e-wT nmin/T2. (2.7)So we haveceff = -6a'miin. (2.8)As a check, note that in the case of the type 0 theory, the lightest state is a tachyonCV'mi = -2. This corresponds to celf = 12, the correct value once one subtracts the 2dimensions cancelled out by the ghosts. Similarly in the bosonic string, the lightest stateis a tachyon with a'mti = -4, corresponding to celf = 24.For models in which the mass squared is nonnegative, such as spacetime-supersymmetricstrings, there is no such exponential IR divergence in the partition function. Hence, bymodular invariance, there is also no surviving contribution of order eCOnst/T2. This requiresa dramatic cancellation going well beyond the necessary condition that the effective centralcharges of the spacetime bosons and fermions agree, c = ce [5].Theories which do have such IR divergences might naively be expected to universallysuffer from dramatic instabilities on equal footing with those in the bosonic string theory.However, this is not the case; IR divergences can arise from modes, pseudotachyons, whosecondensation does not cause a large back reaction (as occurs in the standard case of infla-tionary perturbation theory, as well as in more formal time-dependent string backgrounds[10,11]). This is the case we will encounter in our examples. In the next section we willstudy the UV and IR limits of the partition function, and the modular transformationbetween them, in the case of spacetimes with compact negatively curved spatial slices.3. Exponential Growth: ExamplesAny compact manifold of negative sectional curvature has a fundamental group ofexponential growth [2], meaning that the number of independent elements grows exponen-tially with minimal word length in any basis of generators of the group. It follows [3] thatthe number of free homotopy classes also grows exponentially with geodesic length.In this section, we reproduce this result in constant curvature examples using modularinvariance in string theory. This analysis will provide precise results for the IR and UVlimits of the partition function, and their modular equivalence, exhibiting these compactnegatively curved target spaces as supercritical backgrounds of string theory as suggestedin [1]. We will then consider more general examples and applications.